Abstract: Strong, long-range interactions between atoms in high-lying Rydberg states
make them attractive systems for the studies of ordered phases of interacting
many-body systems and simulating quantum phase transitions.

Several conceptually different approaches have been explored, both theoretically
and experimentally, for the preparation of crystalline order of Rydberg excitations
in spatially-extended ensembles of cold atoms. These include direct (near-)resonant
laser excitation of strongly-interacting Rydberg states in a two-dimensional lattice
gas, and adiabatic preparation of crystalline phases of Rydberg excitations in a
one-dimensional optical lattice by adiabatic frequency sweep of the excitation laser.
We show, however, that taking into account realistic relaxation processes affecting
the atoms severely complicates the prospects of attaining sizable crystals of Rydberg
excitations in laser-driven atomic media. Our simulations well reproduce the
experimental observations of spatial ordering of Rydberg excitations in driven
dissipative lattice gases, as well as highly sub-Poissonian probability distribution
of the excitation number. We find that the excitations essentially form liquid rather
than crystal phases with long-range order.